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MiS Preprint
7/2022
The geometry of one-weight codes in the sum-rank metric
Alessandro Neri, Paolo Santonastaso and Ferdinando Zullo
We provide a geometric characterization of $k$-dimensional $\mathbb F_{q^m}$ linear sum-rank metric codes as tuples of $\mathbb F_q$-subspaces of $(\mathbb F_{q^m})^k$. We then use this characterization to study one-weight codes in the sum-rank metric. This leads us to extend the family of linearized Reed-Solomon codes in order to obtain a doubly-extended version of them. We prove that these codes are still maximum sum-rank distance (MSRD) codes and, when $k=2$, they are one-weight, as in the Hamming-metric case. We then focus on constant rank profile codes in the sum-rank metric, which are a special family of one weight codes, and derive constraints on their parameters with the aid of an associated Hamming-metric code. Furthermore, we introduce the $n$-simplex codes in the sum-rank metric, which are obtained as the orbit of a Singer subgroup of $\mathrm{GL}(k,q^m)$. They turn out to be constant rank-profile -- and hence one-weight -- and generalize the simplex codes in both the Hamming and the rank metric. Finally, we focus on $2$-dimensional one-weight codes, deriving constraints on the parameters of those which are also MSRD, and we find a new construction of one-weight MSRD codes when $q=2$.
